lect7 - Counting problems for sets Cartesian Product and...

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1 The Rule of Product: If something can happen in n ways, and no matter how the first thing happens, a second thing can happen in m ways, then two things together can happen in n·m ways. Example . How many two-symbol passwords exist, if the first symbol should be a letter and the second – a digit? Let A be the set of letters and B be the set of digits. Then A × B is the set of all passwords satisfied the above restrictions. | A × B |=|A| | B | = 26 10 Cartesian Product and Rule of Product A × B = {( a , b ) | a A and b B } | A × B |=|A| | B | Counting problems for sets
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2 The Rule of Sum If the first event can occur in m ways, the second event can occur in n ways, two events can not occur simultaneously, then either the first event or the second can occur in m + n ways. Translate it for sets: The Rule of Sum (Axiom). If A and B are two (finite) sets and A B = ( disjoint sets) then | A B| = | A | + | B |. Example . If we can chose either letter (set A ) or digit (set B ), then A B = and A B is the set of choices for each symbol. | A B| = | A | + | B | = 26+10, the number of choices.
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3 The rule of Sum can be generalized for n disjoint sets. If A 1 , A 2 , …, A n are n finite sets, such as A i A j = for any i j , 1 i, j n , then | A 1 A 2 A n |=| A 1 | + | A 2 | +…+| A n |. It is certainly consistent with our experience. The condition A B = ensures that no element is counted twice.
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4 Lemma . Let A and B any two sets. Then a) A = ( A - B ) ( A B ) b) ( A - B ) ( A B ) = What if A B ? Theorem 11(Inclusion exclusion principle). | A B| = | A | + | B | - | A B |. A B A - B A B
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5 a) A = ( A - B ) ( A B ) Proof. We need to prove that 1) A ( A - B ) ( A B ) and 2) ( A - B ) ( A B ) A. 1) Take any x A. For this x we have that either x B or x B is true. In the former case x A B by the definition of , and in the later case x A and x B imply that x A - B. Thus, any x A belongs either to A B or to A - B , i. e. x ( A - B ) ( A B ) by the dfn of . 2) Take any
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lect7 - Counting problems for sets Cartesian Product and...

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